Question

In the figure, ▢ABCD is a cyclic quadrilateral. If m(arc ABC) = 230°, then find ∠ABC, ∠CDA, ∠CBE.

Answer 1

m(arc ABC) = 230°   ...(i) [Given]

∴ m(arc ADC) + m(arc ABC) = 360°   ...[Degree measure of a circle is 360°]

∴ m(arc ADC) = 360° – m(arc ABC)

∴ m(arc ADC) = 360° – 230°   ...[From (i)]

∴ m(arc ADC) = 130°

`∠ABC = 1/2 m(arc  ADC)`   ...[Inscribed angle theorem]

= `1/2 xx 130^circ`

= 65°

Now, `∠CDA = 1/2 m(arc  ABC)`   ...[Inscribed angle theorem]

∴ `∠CDA = 1/2 xx 230^circ`

∴ ∠CDA = 115°   ...(ii)

∠CBE = ∠CDA   ...(iii) [The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle]

∴ ∠CBE = 115°   ...[From (ii) and (iii)]

∴ ∠ABC = 65°, ∠CDA = 115°, ∠CBE = 115°.